The Bethe Ansatz

Let us look more closely at our retarded correlation function Cret. Setting \(t' = 0\) without loss of generality, and expliciting the interaction representation for the operators, we get

\[ {\cal C}^{\hat{O}, \hat{P}}_{ret, \psi_\alpha} (t) = -i \theta(t) \langle \psi_\alpha | \left(e^{i H_0 t} \hat{O} e^{-i H_0 t} \hat{P} - \hat{P} e^{i H_0 t} \hat{O} e^{-i H_0 t} \right) |\psi_\alpha \rangle. \]

Introducing a resolution of the identity \({\bf 1} = \sum_{\alpha} |\psi_\alpha \rangle \langle \psi_\alpha|\) between the two operators and defining the matrix elements

\[ \langle \psi_{\alpha} | A | \psi_{\alpha'} \rangle \equiv A_{\alpha \alpha'} \]

we can write

\[ {\cal C}^{\hat{O}, \hat{P}}_{ret, \psi_\alpha} (t) = -i \theta (t) \sum_{\alpha'} \left(O_{\alpha \alpha'} P_{\alpha' \alpha} e^{i (E_\alpha - E_{\alpha'}) t} - P_{\alpha \alpha'} O_{\alpha' \alpha} e^{-i(E_\alpha - E_{\alpha'}) t} \right). \]

Let us now introduce the Fourier transform (in time) of this retarded correlation function

\[ {\cal C}^{\hat{O}, \hat{P}}_{ret, \psi_\alpha} (\omega) \equiv \int_{-\infty}^\infty dt ~{\cal C}^{\hat{O}, \hat{P}}_{ret, \psi_\alpha} (t) e^{i\omega t - \eta |t|} \]

in which we have introduced a regulator \(\eta \rightarrow 0^+\) to ensure that the time integral converges. Performing this transform explicitly gives

\[ {\cal C}^{\hat{O}, \hat{P}}_{ret, \psi_\alpha} (\omega) = \sum_{\alpha'} \left( \frac{O_{\alpha \alpha'} P_{\alpha' \alpha}}{\omega + E_\alpha - E_{\alpha'} + i\eta} - \frac{P_{\alpha \alpha'} O_{\alpha' \alpha}}{\omega - (E_\alpha - E_{\alpha'}) + i\eta} \right). \]

A representation such as this, where the time dependence has been explicitly extracted by using an eigenstate basis, is known as a Lehmann representation. Viewed as a function of the real frequency \(\omega\) extended to complex values \(\omega \in \mathbb{C}\), this retarded correlation function has singularities in (real) frequencies, which are at positions \(\pm (E_\alpha - E_\beta) - i\eta\), in other words which are exclusively in the lower half-plane of \(\omega\). Any retarded correlation function is thus analytic in the upper half-plane of such a (generically complex-valued) \(\omega\).

Note that introducing \(\eta \rightarrow 0\) is purely a convenience trick to make calculations easily tractable. Without it, we would for example have to deal with integrals of the form

\[ \int_{-\infty}^\infty dt \theta (t) e^{i(\omega - E)t} = \int_0^\infty dt e^{i(\omega - E)t} \]

which are ill-defined as they stand. On the other hand, with the regulator, they become trivial:

\[ -i \int_0^\infty dt e^{i(\omega - E)t - \eta t} = \frac{1}{\omega - E + i\eta}. \]

The physical interpretation which can be given to the regulator is that it represents generic decoherence between wavefunctions at large times: quantum oscillations between states don't remain phase coherent forever, leading to the eventual decay (in time) of correlations. To separate the real and imaginary parts of a correlator, we can make use of a particularly useful identity due to Dirac:

\[ \lim_{\eta \rightarrow 0^+} \frac{1}{\omega \pm i\eta} = \mp i\pi \delta (\omega) + P \frac{1}{\omega} \]

in which \(P\) represents taking the principal part of the integral in which this function stands.